Functor Cohomology. Vincent Franjou. Laboratoire Jean-Leray, Université de Nantes. October 21st, VF (LMJL) Functor cohomology DCAT / 15

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1 Functor Cohomology Vincent Franjou Laboratoire Jean-Leray, Université de Nantes October 21st, 2007 VF (LMJL) Functor cohomology DCAT / 15 Invariant theory Let K be a field, V a dimension n vector space, and let G be a subgroup of GL(V ). Let G act on K[V ] := K[x 1,..., x n ] by linear substitution. Theorem (Hilbert 1890) Suppose that the representation of G on the coordinate ring K[V ] = K[x 1,..., x n ] is semi-simple. Then the ring of invariants K[V ] G is a finitely generated K-algebra. Problem (Hilbert s 14th) Let F be a subfield of the field of fractions K(V ) := K(x 1,..., x n ) which contains K. Is K[V ] F a finitely generated K-algebra? VF (LMJL) Functor cohomology DCAT / 15

2 Hilbert s 14th problem In 1958, Nagata found a counter-example to Hilbert s 14th problem. In 1960, he characterized the groups G such that, whenever G acts on a finitely generated K-algebra, the ring of invariants A G is a finitely generated K-algebra as well. Remark: All this still makes sense when G is an affine group scheme over K, that is a representable functor from finitely generated commutative K-algebras to groups or equivalently the representing finitely generated commutative Hopf algebra K[G], whose dual is the co-commutative group algebra KG. Nagata s property is referred to as geometric reductivity. VF (LMJL) Functor cohomology DCAT / 15 Higher invariant theory The same type of question has been asked for the cohomology ring H (G, A). Here we consider the cohomology defined by Hochschild, the cohomology of the cocommutative Hopf algebra KG. This is group cohomology for a discrete group, rational cohomology for a linear algebraic group. Problem (van der Kallen) Whenever G acts on a finitely generated K-algebra A, is the cohomology ring H (G, A) a finitely generated K-algebra as well? VF (LMJL) Functor cohomology DCAT / 15

3 Higher invariant theory Positive answers have been given when G is a finite group [Evens, 1961] KG is a f. d. graded connected cocommutative Hopf algebra [Wilkerson, 1981] KG is the enveloping algebra of a f. d. restricted Lie algebra [Friedlander & Parshall, 1983] KG is finite-dimensional [Friedlander & Suslin, 1997]. In a series of recent papers, van der Kallen argues that the answer should be positive always, as soon as it is positive for the ring of invariants A G = H 0 (G, A), that is under Nagata s geometric reductivity. VF (LMJL) Functor cohomology DCAT / 15 Higher invariant theory Proofs reduce to the case of the general linear group GL n and use spectral sequences. Typically, Friedlander and Suslin use May s spectral sequence for the Frobenius kernels GL n(r) : E 2s,t 0 = r i=1 S s (gl (i)# n ) r i=1 Λ t (gl (i 1)# n ) A = H 2s+t (GL n(r), A) where gl n = Lie(GL n ). What is needed at this level is enough permanent classes to generate a subalgebra which E 2 is a finite module over. This is an instance where functor cohomology has proven useful. VF (LMJL) Functor cohomology DCAT / 15

4 Polynomial functors (after Friedlander & Suslin, Pirashvili) We fix a positive integer d and describe what an homogeneous polynomial functor of degree d is [Pirashvili, P & S 2003]. We start with the d-th divided power functor of a K-vector space V : Γ d (V ) := (V d ) S d Let V be the (small) category of f. d. K-vector spaces. Definition The category Γ d V has same objects as V and has morphisms Hom Γ d V(V, W ) := Γ d (Hom K (V, W )). Note that Γ d is indeed defined over Z, and that Γ d V is just the K-linearization of Γ d (ab), where ab is the (small) category of finite rank free abelian groups. VF (LMJL) Functor cohomology DCAT / 15 Polynomial functors Definition An homogeneous polynomial functor of degree d is a K-linear functor Γ d V V. We let P d be the category of (natural transformations between) homogeneous polynomial functors of degree d. That is: the structural map is an homogeneous degree d polynomial. Examples the d-th tensor power functor d ; the d-th divided power functor Γ d ; for any f. d. representation M of S d, Hom Sd (M, d ). Write I := Γ 1 = 1 for the identity functor. VF (LMJL) Functor cohomology DCAT / 15

5 Polynomial functors The category P d enjoys many nice properties allowing homological computations. Polynomial functors admit base change. This is more easily seen if describing P d as the category of additive functors from Γ d (ab) to V. The functor P n := Γ d (Hom(K n, )) is a projective generator for n d. Note that End Pd (P n ) = Γ d (End(K n )) = K[M n ] # d is the Schur algebra, so that P d is equivalent to the category of f. d. degree d polynomial representations of GL n for n d. Thus, evaluation yields Ext P d (F, G) H (GL n, Hom(F (K n ), G(K n ))) which is an isomorphism when n > d. VF (LMJL) Functor cohomology DCAT / 15 Cohomology computations (last century) Most computations stem from the 1994 paper F-Lannes-Schwartz - they are just easier in P. Now K is a characteristic p finite field. For a K-vector space V, V (1) is the K-vector space obtained by extension of scalars through the Frobenius. For a functor F in P d, F (1) in P pd is defined by: F (1) (V ) := F (V (1) ). Theorem E r := Ext P (I (r), I (r) ) is a divided power algebra on a generator e 0 in degree 2 truncated at height r. Thus, it is concentrated in even degrees up to 2(p r 1) and one-dimensional in those.... This has been widely extended in [FFSS, Annals Math. 1999], which provided the basic computations for Cha lupnik (Annales Sc. ENS 2005). VF (LMJL) Functor cohomology DCAT / 15

6 Cohomology computations (after Cha lupnik) Cha lupnik notices that functors in P d can always be written F = f ( d ) for a functor f from f. d. representations of S d to V. Indeed there are two natural choices, the left and the right adjoint of the functor f F = f ( d ). A representable choice f = Hom Sd (M, ) is not always possible, but both adjoints take values among coherent functors (F-Pirashvili, preprint). V S d mod j P d j! j VF (LMJL) Functor cohomology DCAT / 15 Cohomology computations (after Cha lupnik) Cha lupnik then proves formulae expressing Ext P (F (r) dp r, G (r) ) in terms of f, g and the graded S op d S d-representation T r := Ext P dp r ( d(r), d(r) ) = E d r KS d. In many cases, for instance for a projective F, it is given by: As a consequence: Ext P dp r (F (r), G (r) ) = g f # (T r ). Ext P dp r (Γ d(r), G (r) ) = G(Ext P p r (I (r), I (r) )). VF (LMJL) Functor cohomology DCAT / 15

7 Cohomology computations (after Cha lupnik) Resolving a general F by projectives, we get from the resulting spectral sequence: Theorem (F-Pirashvili) There is a natural graded K-linear functor e(f, G) defined on S op d S d- modules such that e(f, G)(T r ) is the second page of a spectral sequence converging to Ext P (F (r), G (r) ), for all polynomial functors F, G of deg d. Note that collapsing gives a functor-proof of twist injectivity, hence, combined with [Kuhn, 1995, p ], a functor-proof of Ext-injectivity through precomposition Ext P d (F, G) Ext P da (F A, G A). VF (LMJL) Functor cohomology DCAT / 15 Bifunctor cohomology In van der Kallen s finite generation problem, desirable classes live in H(GL n, Γ d gl n ). This can be expressed in terms of functor cohomology if one allows two-variable functors. Define: gl(v, W ) := Hom(V, W ). Proposition (F-Friedlander 2007) Let P(1,1) be the category of strict polynomial functors defined on V op V and let B in P(1,1) be of bidegree (d, d). For n d, there is an isomorphism: Ext P(1,1) (Γ d gl, B) = H(GL n, B(K n, K n )). We denote by H(GL, B) this stable cohomology. The one variable case is recovered by: Ext P (F, G) = Ext P(1,1) (Γ d gl, gl (F, G)). VF (LMJL) Functor cohomology DCAT / 15

8 Touzé s cohomology computations Theorem (A. Touzé 2007) The Poincaré series of H P (GL, S d(r) gl) is equal to the Poincaré series of the coinvariants of T r = H P (GL, d(r) gl) under the action of the symmetric group S d. Touzé s methods don t give yet an answer for H(GL, Γ d(r) gl). However, he obtains several relevant results, as the top cohomological degree, or: Theorem (A. Touzé 2007) The Euler-Poincaré characteristic of H P (GL, F gl) and of H P (GL, F gl) are equal. Here F # is the Kuhn dual of F, for example: S d# = Γ d. VF (LMJL) Functor cohomology DCAT / 15